DIRECT PRODUCTS OF QUASIREGULAR MAPPINGS ON METRIC SPACES

ON METRIC SPACES

MIHAI CRISTEA

We show that a direct productF = f×g of quasiregular mappings on metric spaces is quasiregular if and only if f and g are L-BLD maps and satisfy the compatibility conditions(∗)or(∗∗).

AMS 2000 Subject Classication: 30C65.

Key words: quasiregular map, metric space.

1. INTRODUCTION AND PRELIMINARIES

Throughout this paper we consider only metric spaces. With an obvious abuse of the notation concerning the metrics involved, we take on the product spaceX×Y the distanced((a, b),(α, β)) = max{d(a, α), d(β, b)}. Iff :X→Y is a map, x∈X andr >0, for 0< r < ρ we dene

L(x, f, r) = sup

d(x,y)=r

d(f(x), f(y)), l(x, f, r) = inf

d(x,y)=rd(f(x), f(y)), L(x, f, r) =e sup

d(x,y)≤r

d(f(x), f(y)), ˜l(x, f, r, ρ) = inf

r≤d(x,y)≤ρd(f(x), f(y)).

If X is locally compact and f is continuous, we can nd y, z ∈ S(x, r) such that L(x, f, r) =d(f(x), f(y)),l(x, f, r) =d(f(x), f(z)) and we can nd w∈B(x, r)¯ andu∈B(x, ρ)\B(x, r)¯ such thatL(x, f, r) =˜ d(f(x), f((w))and

˜l(x, f, r, ρ) =d(f(x), f(u)). Iffis discrete atxandB(x, ρ¯ 0)∩f⁻¹(f(x)) ={x}, then l(x, f, r) > 0 for 0 < r ≤ ρ₀. Also, if 0 < ρ₁ < ρ₂ ≤ ρ₀ and λ =

inf

ρ1≤d(x,y)≤ρ2

d(f(x), f(y))>0, then lim

r→0l(x, f, r) = 0, hence there exists r0 >0 such that l(x, f, r) < λ for 0 < r ≤ r0, and this implies that ˜l(x, f, r, ρ1) =

˜l(x, f, r, ρ2)for 0< r≤r0.

For a continuous, discrete mapping f : X → Y dened on a locally compact space, we can therefore correctly dene the linear dilatations

H(x, f) = lim sup

r→0

L(x, f, r)

l(x, f, r), H(x, f˜ ) = lim sup

r→0

L(x, f, r)˜

˜l(x, f, r, ρ)

REV. ROUMAINE MATH. PURES APPL., 53 (2008), 4, 285296

of f inx, and we see thatH(x, f)˜ does not depend on ρ≤ρ0.

If X, Y are domains from Rⁿ, a homeomorphism f :X →Y for which there exists H ≥1 such thatH(x, f) ≤H for every x∈X is called quasicon- formal. An extensive literature was devoted to this subject in the last 70 years (see for instance [1] and [12]). Karmazin [5] proved that if f and gare homeo- morphisms between some domains fromRⁿ, thenF =f×gis quasiconformal if and only if the compatibility conditions

(∗)

H(x, y, f, g) = lim sup

r→0

L(x, f, r) l(y, g, r) ≤H, H(x, y, g, f) = lim sup

r→0

L(y, g, r) l(x, f, r) ≤H hold on X×Y. We also introduce the compatibility conditions

(∗∗)

H(x, y, f, g) = lim sup˜

r→0

L(x, f, r)˜

˜l(y, g, r, ρ) ≤H, H(x, y, g, f˜ ) = lim sup

r→0

L(y, g, r)˜

˜l(x, f, r, ρ) ≤H.

Quasiregular mappings were considered in 1966 by Reshetnyak ([8], [9]) and a few years later by Martio, Rickman, and Väisälä [7]. A mappingf :D→ Rⁿ is quasiregular if D⊂Rⁿ is open, f is ACLⁿ and there exists K ≥1 such that kf⁰(x)kⁿ ≤ K·J_f(x) a.e.. It is proved in [7] that f is quasiregular and nonconstant if and only if f is open, and discrete, H(·, f) is locally bounded on D and H(·, f) is bounded on D\B_f, where B_f = {x ∈ D | f is not a local homeomorphism at x} is a set such thatµ_n(B_f) = 0. Elena Rusu [11]

proved that ifX,Y are domains inRⁿ andR^m,f :X →Rⁿ,g:Y →R^m are continuous, open, discrete and sense-preserving, then F =f ×g is such that H((x, y), F)≤H for any(x, y)∈X×Y and some H≥1, if and only iff and g satises the compatibility conditions (∗).

Martio and Väisälä [6] considered the class of L-BLD mappings (with bounded length distortion), which are mappings f :D→ Rⁿ, with D⊂Rⁿ a domain, such that one of the following equivalent condition is satised:

i)f is ACL,khk/L≤ kf⁰(x)(h)k ≤Lkhk for allh ∈Rⁿ and a.e. x∈D, and J_f(x)≥0 a.e.

ii) For every x∈D, there exists rx >0 such that ky−xk/L≤ kf(y)− f(x)k ≤Lky−xkfor every y∈B(x, r¯ _x) and J_f(x)≥0. a.e.

iii)f is open, discrete, sense-preserving and l(α)/L≤l(f ◦α)≤L·l(α) for every path α inD.

iv) L(x, f)≤L,l(x, f)≥ _L¹ for everyx∈D, andJ_f(x)≥0 a.e.

v)f is quasiregular and kf⁰(x)k ≤L,l(f⁰(x))≥ _L¹ a.e.

Here l(α) is the length of the path α, kAk = sup

khk=1

kA(h)k, l(A) =

khk=1inf kA(h)k for A ∈ L(Rⁿ,Rⁿ) and L(x, f) = lim sup

y→x

kf(y)−f(x)k

ky−xk , l(x, f) = lim inf

y→x

kf(y)−f(x)k ky−xk .

IfX, Y are metric spaces, we say that a mapingf :X→Y isL-BLD at the point x ∈X if there exists r_x > 0 such that d(x, y)/L ≤d(f(x), f(y))≤ L d(x, y) for every y ∈ B(x, rx), and we say that f is L-BLD if it is L-BLD at every point x ∈ X. We see that this denition is more general than that given by Martio and Väisälä [6] even for mappings dened between Euclidean domains, since we don't ask f to be such thatJ_f(x)≥0 a.e..

In the last years there were also considered homeomorphismsf :X→Y between metric spaces. Such mapings are called quasiconformal if the linear dilatation H(·, f) or H(·e , f) are uniformly bounded on X (see the book of Heinonen [4]). We started the research concerning quasiregular mappings on metric spaces in [3]. We call quasiregular a continuous, open, discrete mapping f :X →Y between two metric spaces such that the linear dilatation H(·, f) is uniformly bounded on X. We see that this denition coincide with the classical one in the Euclidean case for mappings with N(f, X) < ∞. We proved in [3] in the caseN(f, X)<∞the equivalence of the metric denition of quasiregularity (i.e. the denition concerning the uniform boundedness of the linear dilatation H(·, f)onX) and the geometric denition of quasiregularity.

The central result of the paper is that under some minimal assumptions, the direct productF =f×g of two continuous discrete mappingsf :X→Z, g :Y → W is such that H((x, y), F) ≤ H on X×Y for some H ≥ 1 if and only if f and g are L-BLD mappings for some L > 0 and the compatibility conditions (∗) or (∗∗) are satised for some H > 0 on X ×Y. Our results are new even for mappings between Euclidean domains in Rⁿ, since we also consider the compatibility relations (∗∗) and we consider continuous, discrete and possibly non open mappings for which the linear dilatation H(·, f) may be uniformly bounded (such a mapping is F : C→ C, f(z) =z if Imz ≥ 0, f(z) = ¯z if Imz < 0, which is discrete not open and H(z, f) = 1 for every z∈C). Recently, we proved [2] that such mappings satisfy some basic modular inequalities, extending in this way the class of quasiregular mappings. Also, the proofs of the results from Section 2 are somewhat simpler than the proofs from [5] and [11], since we avoid Markusevic-Pesin's denition of quasiconformality.

Forf :X×Y →Z,x∈X,y∈Y we denote byf_x:Y →Z,f^y :X→Z the mappings dened byfx(y) =f(x, y),f^y(x) =f(x, y),(x, y)∈X×Y. We have the following obvious result.

Lemma 1. Let X be a locally compact space, Y a normed space and f : X → Y open and continuous. Then the mapping r → L(x, f, r) is increasing for every x∈X.

We say thatf :X→Y satises a local maximum principle atx if there existsr_x>0 such that the mappingr→L(x, f, r) is incresing for 0< r < r_x. We see from Lemma 2 that if X is locally compact,Y is a normed space, and f :X→Y is open and continuous, then f satises a local maximum principle at every point x∈X.

2. CALCULUS OF THE LINEAR DILATATION OF THE PRODUCT MAP

We consider a more general case, namely, mappingsF :X×Y →Z×W given by F(x, y) = (f(x), g(x, y)), (x, y)∈ X×Y, and give some bounds for the linear dilatation H((x, y), F). In the special case F(x, y) = (f(x), g(y)), (x, y)∈X×Y, we calculate H((x, y), F).

Theorem 1. LetX, Y be locally compact, letf :X→Z,g:X×Y →W be continuous, and F : X ×Y → Z ×W given by F(x, y) = (f(x), g(x, y)) for (x, y) ∈ X ×Y, x0 ∈ X, y0 ∈ Y such that f is discrete at x0 and gx0

is discrete at y₀. Then F is discrete at (x₀, y₀) and if r₀ > 0 is such that B¯(x0, r0)∩f⁻¹(f(x0)) ={x₀}, B(y¯ 0, r0)∩g_x⁻¹₀(gx0(y0)) ={y₀}, then

(1) max{L(x0, f, r), L(y0, gx₀, r)}

min{l(y0, gx0, r),max{l(x0, f, r), L(x0, g^y0, r)}}≤ L((x0, y0), F, r)

l((x0, y0), F, r), 0< r≤r0,

(2) max{L(x˜ 0, f, r),L(y˜ 0, gx₀, r)}

min{˜l(y0, gx₀, r, r0),max{˜l(x0, f, r, r0),L(x˜ 0, g^y0, r0)}} ≤ L((x˜ 0, y0), F, r)

˜l((x0, y0), F, r0), 0< r≤r0.

Proof. Let 0< r ≤ r0 and let E : ¯B(x0, r0)×B(y¯ 0, r0) → R,E(x, y) = max{d(f(x), f(x0)), d(g(x, y), g(x0, y0))} for (x, y) ∈ B(x¯ 0, r0)×B(y¯ 0, r0). If

¯

x∈S(x₀, r)is such thatL(x₀, f, r) =d(f(x₀), f(¯x))and ify¯∈S(y₀, r)is such that L(y0, gx0, r) = d(gx0(¯y), gx0(y0)), then E(¯x, y0) = max{d(f(¯x), f(x0)), d(g_x¯(y₀), g_x0(y₀))} ≥ d(f(¯x), f(x₀)) = L(x₀, f, r) and E(x₀,y) =¯ d(g_x0(y₀), gx0(¯y)) =L(y0, gx0, r).

SinceL((x₀, y₀), F, r) = sup

max{d(x₀,x),d(y0,y)}=r

E(x, y), we have (3) max{L(x₀, f, r), L(y₀, g_x0, r} ≤L((x₀, y₀), F, r), 0< r≤r₀. In the same manner we nd that

(4) max{L(x˜ 0, f, r),L(y˜ 0, gx0, r)} ≤L((x˜ 0, y0), F, r), 0< r≤r0.

Let now x¯ ∈ S(x0, r), y¯ ∈ S(y0, r0) be such that d(f(x0), f(¯x)) = l(x₀, f, r) and d(g_x0(y₀), g_x0(¯y)) =l(y₀, g_x0, r). Then E(¯x, y₀) = max{d(f(¯x), f(x0)), d(g^y0(¯x), g^y0(x0))} ≤ max{l(x₀, f, r), L(x0, g^y0, r)}, and E(x0,y) =¯ d(gx0(¯y), gx0(y0)) =l(y0, gx0, r).

Sincel((x₀, y₀), F, r) = inf

max{d(x₀,x),d(y0,y)}=rE(x, y), we have (5)

l((x0, y0), F, r)≤min{l(y₀, gx0, r)},max{l(x₀, f, r), L(x0, g^y0, r)}, 0< r≤r₀. Similarly, we prove that

(6)

˜l((x0, y0), F, r, r0)≤min{˜l(y0, gx0, r, r0)}, max{˜l(x0, f, r, r0),L(x˜ 0, g^y0, r0)}, 0< r≤r0.

Using (3) and (5), respectively (4) and (6) we obtain formulae (1) and (2).

Theorem 2. Let X, Y be locally compact, let f :X → Z, g :Y → W be continuous, and F : X×Y → Z ×W, F = f ×g, x0 ∈ X such that f is discrete at x₀, y₀ ∈ Y such that g is discrete at y₀ and let r₀ > 0 be such that B¯(x0, r0)∩f⁻¹(f(x0)) ={x₀}, B(y¯ 0, r0)∩g⁻¹(g(y0)) ={y₀}. Then ( ¯B(x₀, r₀)×B(y¯ ₀, r₀))∩F⁻¹(F(x₀, y₀)) ={(x₀, y₀)} and

max{L(x₀, f, r), L(y₀, g, r)}

min{l(x₀, f, r), l(y0, g, r)} ≤ L((x₀, y₀), F, r)

l((x0, y0), F, r), 0< r≤r₀, (7)

max{L(x˜ ₀, f, r),L(y˜ ₀, g, r)}

min{˜l(x0, f, r, r0),˜l(y0, g, r, r0)} ≤ L((x˜ ₀, y₀), F, r)

˜l((x0, y0), F, r, r0), 0< r≤r₀. (8)

If H((x0, y0), F) ≤ H then H(x0, y0, f, g) ≤ H, H(x0, y0, g, f) ≤ H, H(x0, f)≤H, H(y0, g)≤H, and ifH((x˜ 0, y0), F)≤H then H(x˜ 0, y0, f, g)≤ H,H(x˜ ₀, y₀, g, f)≤H,H(x˜ ₀, f)≤H,H(y˜ ₀, g)≤H.

Proof. We have L(x0, g^y0, r) = 0 for 0 < r ≤ r0 and then apply Theo-

rem 1.

Theorem 3. Let X, Y be locally compact, let f :X→ Z, g :X×Y → W be continuous, and F : X×Y → Z ×W, F(x, y) = (f(x), g(x, y)) for (x, y) ∈X×Y, x₀ ∈X such thatf is discrete at x₀, y₀ ∈Y such thatg_x0 is discrete at y0, and let r0 >0 be such that B¯(x0, r0)∩f⁻¹(f(x0)) ={x₀} and B¯(y₀, r₀)∩g_x⁻¹₀ (g_x0(y₀)) ={y₀}. ThenF is discrete at (x₀, y₀) and

(9) _˜^{L((x˜ 0,y0),F,r)}

l((x0,y0),F,r,r0) ≤

max

n

L(x˜ 0,f,r),L(y˜ 0,gx0,r)+ sup

y∈B(y¯ 0,r)

L(x˜ 0,g^y,r)

o

min{L(x˜ 0,f,r),˜ρ(r)} , 0< r≤r0,

whereρ(r) = inf˜

0<s<rmax

n˜l(x0, f, s, r0),˜l(y0, gx0, r, r0)− sup

y∈B(y¯ 0,r0)

L(x˜ 0, g^y, s) o

>

0,0< r < r0.

If f satises a maximum principle at x₀ and g_x0 satises a maximum principle at y0, we have

(10)

L((x0,y0),F,r) l((x0,y0),F,r) ≤

max

n

L(x0,f,r),L(y0,gx0,r)+ sup

y∈B(y¯ 0,r),0<s≤r

L(x0,g^y,s)

o

min{l(x0,f,r),ρ(r)} , 0< r≤r0, where ρ(r) = inf

0<s<rmaxn

l(x₀, f, s), l(y₀, g_x0, r)− sup

y∈S(y0,r)

L(x₀, g^y, s)o

> 0, 0< r≤r0.

Proof. LetE: ¯B(x₀, r₀)×B(y¯ ₀, r₀)→R,E(x, y) = max{d(f(x), f(x₀)), d(g(x, y), g(x0, y0))} for (x, y)∈B(x¯ 0, r0)×B¯(y0, r0) and let0 < r≤r0. Let (x, y) ∈ S((x₀, y₀), r) be such that L((x₀, y₀), F, r) = E(x, y). If d(x₀, x) =r and d(y0, y) = s < r, then L((x0, y0), F, r) = E(x, y) = max{d(f(x), f(x0)), d(g(x, y), g(x₀, y₀))} ≤max{L(x₀, f, r), d(g_x0(y), g_x0(y₀))+d(g^y(x), g^y(x₀))} ≤ maxn

L(x₀, f, r), L(y₀, g_x0, r) + sup

y∈B(y¯ 0,r),0<s≤r

L(x₀, g^y, s)o

. Ifd(y, y₀) =rand d(x0, x) =s < r, then

L((x0, y0), F, r) =E(x, y) = max{d(f(x), f(x0)), d(g(x, y), g(x0, y0))}

≤max{L(x₀, f, s), d(g_x0(y), g_x0(y₀)) +d(g^y(x), g^y(x₀))}

≤max n

L(x0, f, r), L(y0, gx0, r) + sup

y∈B(y¯ 0,r),0<s≤r

L(x0, g^y, s) o

. So, we proved that

(11) L((x0, y0), F, r)≤max{L(x₀, f, r), L(y0, gx0, r)

+ sup

y∈B(y¯ 0,r),0<s<r

L(x0, g^y, s)} for 0< r≤r0.

Let now(x, y) ∈S((x0, y0), r) be such that l((x0, y0), F, r) = E(x, y). If d(x₀, x) = r then l((x₀, y₀), F, r) = E(x, y) = max{d(f(x), f(x₀)), d(g(x, y), g(x0, y0))} ≥ d(f(x), f(x0)) ≥ l(x0, f, r), and if d(y0, y) = r and d(x0, x) = s < r then

l((x₀, y₀), F, r) =E(x, y) = max{d(f(x₀, f(x))), d(g(x, y), g(x₀, y₀))}

≥max{l(x₀, f, s), d(gx0(y), gx0(y0))−d(g^y(x), g^y(x0))}

≥maxn

l(x₀, f, s), l(y₀, g_x0, r)− sup

y∈S(y₀,r)

L(x₀, g^y, s)≥ρ(r)o . It follows that

(12) l((x0, y0), F, r)≥min{l(x₀, f, r), ρ(r)} for0< r≤r0.

Using (11) and (12) we obtain (10). Since the maps r → L(x˜ 0, f, r) and r →L(ye ₀, g_x0, r)are incresing, by the same method we nd that

(13) L((x˜ ₀, y₀), F, r)≤max{L(x˜ ₀, f, r),L(y˜ ₀, g^y, r)

+ sup

y∈B(y¯ 0,r)

L(x˜ 0, g^y, r)}, 0< r≤r0,

(14) ˜l((x₀, y₀, F, r, r₀)≥min{˜l(x₀, f, r, r₀),ρ(r)},˜ 0< r≤r₀. Using (13) and (14) we obtain (9).

Theorem 4. Let X, Y be locally compact, let f : X → Z, g : Y → W be continuous, F : X ×Y → Z × W, F = f ×g, x0 ∈ X such that f is discrete at x₀, y₀ ∈ Y such that g is discrete at y₀, and let r₀ > 0 be such that B(x¯ 0, r0)∩f⁻¹(f(x0)) ={x₀}, B(y¯ 0, r0)∩g⁻¹(g(y0)) ={y₀}. Then ( ¯B(x₀, r₀)×B(y¯ ₀, r₀))∩F⁻¹(F(x₀, y₀)) ={(x₀, y₀)} and

(15) L((x˜ ₀, y₀), F, r)

˜l((x0, y0), F, r, r0) = max{L(x˜ ₀, f, r),L(y˜ ₀, g, r)}

min{˜l(x0, f, r, r0),˜l(y0, g, r, r0)}, 0< r≤r₀, and if H(x˜ 0, y0, f, g)≤H, H(x˜ 0, y0, g, f)≤H, thenH((x˜ 0, y0), F)≤H².

Iff satises a maximum principle atx₀ andg satises a maximum prin- ciple at y₀, then

(16) L((x0, y0), F, r)

l((x₀, y₀), F, r) = max{L(x₀, f, r), L(y0, g, r)}

min{l(x₀, f, r), l(y₀, g, r)} , 0< r≤r0, and if H(x0, y0, f, g)≤H, H(x0, y0, g, f)≤H, thenH((x0, y0), F)≤H².

Proof. Since g does not depend on x, we have L(x, g^y, s) = 0 for ev- ery (x, y) ∈ X ×Y and every 0 < s ≤ r₀. Then ρ(r) ≥ l(y₀, g, r), hence l((x0, y0), F, r)≥min{l(x₀, f, r), l(y0, g, r)}, and by Theorem 3,

(17) L((x₀, y₀), F, r)

l((x₀, y₀), F, r) ≤ max{L(x₀, f, r), L(y₀, g, r)}

min{l(x₀, f, r), l(y₀, g, r)} , 0< r≤r₀.

Using Theorem 2, (7) and (8) we obtain (16). In the same maner we see thatρ(r)˜ ≥˜l(y₀, g, r, r₀), hence˜l((x₀, y₀), F, r, r₀)≥min{˜l(x₀, f, r, r₀),˜l(y₀, g, r, r₀)}

and, by Theorem 3, (18) L((x˜ 0, y0), F, r)

˜l((x₀, y₀), F, r, r₀) ≤ max{L(x˜ 0, f, r),L(y˜ 0, g, r)}

min{˜l(x₀, f, r, r₀),˜l(y₀, g, r, r₀)}, 0< r≤r0. Using (8) and (18), we obtain (15).

Finally ifH(x0, y0, f, g) ≤H, H(x0, y0, g, f) ≤H, then forr >0 small enough we have

L(x₀, f, r)≤Hl(y₀, g, r)≤HL(y₀, g, r)≤H²l(x₀, f, r), L(y0, g, r)≤Hl(x0, f, r)≤HL(x0, f, r)≤H²l(y0, g, r), and using (2) we obtainH((x₀, y₀), F)≤H².

We now give the following generalization of the results of Karmazin [5]

and E. Rusu [11].

Theorem 5. LetX, Y be locally compact, let f :X→Z, g:Y →W be continuous and discrete, and let F : X×Y → Z ×W, F =f ×g. Then F is continuous and discrete, and the linear dilatation H((x, y), F˜ ) is uniformly bounded on X×Y by a constant H ≥1 if and only if the compatibility condi- tions H(x, y, f, g)˜ ≤H˜,H(x, y, g, f˜ )≤H˜ hold onX×Y for some constantH˜. If f satises a maximum principle at every point x∈X and gsatises a max- imum principle at every point y ∈ Y, then the linear dilatation H((x, y), F) is uniformly bounded on X×Y by a constant H if and only if the compatibil- ity conditions H(x, y, f, g) ≤ H˜, H(x, y, g, f) ≤ H˜ hold on X ×Y for some constant H.˜

Remark 1. If the compatibility conditionsH(x, y, f, g)≤H,H(x, y, g, f)

≤ H hold, then H(x, f) ≤ H², H(y, g) ≤ H². Also, if H(x, y, f, g)˜ ≤ H, H(x, y, g, f˜ )≤H, thenH(x, f˜ )≤H²,H(y, g)˜ ≤H².

3. DIRECT PRODUCTS OF L-BLD MAPPINGS

Theorem 6. Let f : X → Z, g : Y → W, F : X ×Y → Z ×W, F =f×g, (x0, y0)∈X×Y. Then F isL-BLD at (x0, y0) if and only if f is L-BLD at x₀ and g isL-BLD at y₀.

Proof. Suppose that F is L-BLD at (x0, y0). Then there exists r0 > 0 such that

(19) d((x, y),(x₀, y₀)/L≤d(F(x, y), F(x₀, y₀)≤Ld((x, y),(x₀, y₀)) for d((x, y),(x0, y0))≤ r0. We take rst y = y0 and then x = x0 in (1) and obtain

(20) d(x, x₀)/L≤d(f(x), f(x₀))≤Ld(x, x₀) ford(x, x₀)≤r₀, (21) d(y, y₀)/L≤d(g(y), g(y₀))≤Ld(y, y₀) ford(y, y₀)≤r₀, hence f isL-BLD at x0 and g isL-BLD at y0.

Suppose now that f is L-BLD at x₀ and g is L-BLD at y₀. Then (20) and (21) hold for some r0 > 0. Let now (x, y) ∈ X × Y be such

that d((x, y),(x0, y0)) = max{d(x, x₀), d(y, y0)} ≤ r0 and let us x such an (x, y). Ifmax{d(x₀, x), d(y₀, y)}=d(x, x₀)thend((x, y),(x₀, y₀)) =d(x, x₀)≤ L d(f(x), f(x0)) ≤ Lmax{d(f(x), f(x0)), d(g(y), g(y0))} = L d((f(x), g(y)), (f(x0), g(y0)) = L d(F(x, y), F(x0, y0)), and in the same manner we show that ifmax{d(x₀, x), d(y₀, y)}=d(y₀, y)thend(((x, y),(x₀, y₀))≤L d(F(x, y), F(x0, y0)). It follows that

(22) d((x, y),(x₀, y₀))/L≤d(F(x, y), F(x₀, y₀)).

Suppose now that max{d(f(x), f(x0)), d(g(y), g(y0))} = d(f(x0), f(x)). Thend(F(x, y), F(x₀, y₀)) =d((f(x), g(y)),(f(x₀, g(g₀))) = max{d(f(x), f(x₀)), d(g(y0), g(y))} = d(f(x0), f(x)) ≤ L d(x0, x) ≤ Lmax{d(x₀, x), d(y0, y)} = L d((x, y),(x₀, y₀)). We so obtained

(23) d(F(x, y), F(x0, y0))≤L d((x, y),(x0, y0)).

We see now from (22) and (23) that F isL-BLD at (x₀, y₀).

Theorem 7. Let f :X×Y, x₀ ∈ X, and suppose that f is L-BLD at x₀. Then H(x₀, f)≤L² andH(x˜ ₀, f)≤L².

Proof. Letr0>0be such thatd(x0, x)/L≤d(f(x), f(x0))≤L d(x₀, x)for d(x0, x)≤r0. Then r/L≤l(x0, f, r)≤L(x0, f, r)≤L r for 0< r≤r0, hence

L(x0,f,r)

l(x0,f,r) ≤ _r/L^L·r =L² for 0< r≤r0. This implies thatH(x0, f)≤L². Theorem 8. Let f : X → Z, g : Y → W, F : X ×Y → Z ×W, F = f ×g, x0 ∈ X, y0 ∈ Y such that f is L-BLD at x0, g is L-BLD at y₀. Then H(x₀, y₀, f, g) ≤ L², H(x₀, y₀, g, f) ≤ L², H(x˜ ₀, y₀, f, g) ≤ L², H(x˜ ₀, y₀, g, f)≤L², H((x₀, y₀), F)≤L², andH((x˜ ₀, y₀), F)≤L².

Proof. Letr₀ >0 be such that d(x₀, x)/L ≤d(f(x), f(x₀))≤L d(x₀, x) for d(x0, x)≤r0 andd(y0, y)/L≤d(g(y), g(y0))≤L d(y, y0) for d(y0, y)≤r0. Then L(x0, f, r) ≤ Lr and _L^r ≤ l(y0, g, r), hence ^L(x_l(y0⁰_,g,r)^,f,r) ≤ _r/L^{L r} = L² for 0 < r ≤ r0. This implies that H(x0, y0, f, g) ≤ L². In the same manner we show that H(x₀, y₀, g, f) ≤L²,H(xe ₀, y₀, f, g) ≤L², H(xe ₀, y₀, g, f) ≤L². By Theorem 6,F isL-BLD at(x₀, y₀)and by Theorem 7 we haveH((x₀, y₀), F)≤ L² and H((x˜ ₀, y₀), F)≤L².

Theorem 9. Let f :X →Z, g:Y →W be L-BLD mappings, F :X× Y →Z×W,F =f×g. ThenF isL-BLD,H((x, y), F)≤L²,H((x, y), Fe )≤ L², H(x, y, f, g) ≤ L², H(x, y, g, f) ≤ L², H(x, y, f, g)e ≤ L², H(x, y, g, f˜ ) ≤ L² for every(x, y)∈X×Y.

Theorem 10. Let f :X → Z, g : Y → W be continuous and discrete such that there existsH >0 such that eitherH(x, y, f, g)≤H,H(x, y, g, f)≤ H on X×Y, orH(x, y, f, g)˜ ≤H,H(x, y, g, f˜ )≤H on X×Y. Suppose that

there exist x1, x2∈X andy1, y2 ∈Y such that

(24) lim inf

r→0

L(y1, g, r)

r >0 and lim sup

r→0

l(y2, g, r) r <∞, (25) lim inf

r→0

L(x₁, f, r)

r >0 and lim sup

r→0

l(x₂, f, r) r <∞.

Then there exists L >0 such that f andg are L-BLD mappings.

Proof. Suppose that lim inf

r→0

L(y1,g,r)

r > m >0. Then sup

r>0 0<s<rinf

L(y1,g,s) s >

m, hence there exists ρ_k → 0 such that inf

0<s<ρ_k

L(y1,g,s)

s > m for every k∈ N, somr≤L(y₁, g, r) for every0< r≤ρ_k and everyk∈N.

Let us show that there exists l >0 such thatlim inf

r→0

l(x,f,r)

r ≥l for every x ∈ X. Indeed, if this does not hold, then, we can nd xp ∈ X such that lim inf

r→0

l(xp,f,r)

r < ¹_p for every p ∈ N. Let us x p ∈ N. We can nd r_k < ρ_k such that l(xp, f, rk) ≤ rk/p for k ∈ N and have p·m = ^{m r}_r ^k

k/p ≤ ^L(y_l(x^1,g,rk)

p,f,rk). Letting k→ ∞ we nd that pm≤H(xp, y1, g, f)≤H. Letting now p → ∞, we reach a contradiction. Let now x ∈ X. There exists r_x > 0 such that

0<r≤rinf x

l(x,f,r)

r ≥ l, hence l·r ≤ l(x, f, r) for 0 < r ≤ r_x. If z ∈ B(x, r_x), d(x, z) =r ≤r_x, thenl·d(x, z) =l·r ≤l(x, f, r)≤d(f(x), f(z)), hence (26) l·d(x, z)≤d(f(x), f(z)) for every z∈B(x, rx).

Suppose now that lim sup

r→0

l(y2,g,r)

r < m < ∞. Then inf

r>0 sup

0<s<r

l(y2,g,s) s <

m, hence there existsρ_k→0 such that sup

0<s<ρk

l(y2,g,s)

s < mfor everyk∈N, so l(y2, g, r)≤m·r for every0< r≤ρk and everyk∈N.

Let us show that there exists L > 0 such that lim sup

r→0

L(x,f,r)

r < L for everyx∈X. Indeed, if this does not hold, then, we can ndx_p ∈Xsuch that lim sup

r→0

L(xp,f,r)

r ≥pfor everyp∈N. Fixp∈N. We can ndrk< ρksuch that p·rk ≤L(xp, f, rk) for every k∈N. We have _m^p = _{m r}^{p rk}

k ≤ ^L(x_l(y^p,f,rk)

2,g,rk). Letting k → ∞ we nd that _m^p ≤ H(xp, y2, f, g) ≤ H. Letting p → ∞, we reach a contradiction. Let now x∈X. There exists rx>0such that sup

0<r≤rx

L(x,f,r) r <

L, hence L(x, f, r) ≤ L r for every 0 < r ≤ r_x. Let z ∈ B(x, r_x) and r = d(x, z) ≤ r_x. We have d(f(x), f(z)) ≤ L(x, f, r) ≤ L r = L d(x, z). So, we proved that

(27) d(f(x), f(z)≤L d(x, z)

for every x∈B(x, rx). Using (24) and (25) we see that f isL-BLD for some L > 0. In the same manner we can prove that g isL-BLD. We use the same argument if H(x, y, f, g)˜ ≤H,H(x, y, g, f˜ )≤H on X×Y.

We now obtain the main result of the paper.

Theorem 11. Let X, Y be locally compact, let f :X →Z, g :Y →W be continuous and discrete, F :X×Y → Z×W, F =f ×g such that there exist x₁, x₂ ∈X and y₁, y₂ ∈Y such that

(28) lim inf

r→0

L(y1, g, r)

r >0 and lim sup

r→0

l(y2, g, r) r <∞,

(29) lim inf

r→0

L(x1, g, r)

r >0 and lim sup

r→0

l(x2, g, r) r <∞.

Thenf andgareL-BLD mappings if and only if there existsH ≥1such that H((x, y), F)≤H on X×Y or H((x, y), F˜ )≤H on X×Y.

Proof. IffandgareL-BLD mappings, by Theorem 9 we haveH((x, y), F)

≤ L² on X×Y and H((x, y), F˜ ) ≤ L² on X ×Y. Suppose now that there exists H ≥ 1 such that H((x, y), F) ≤ H on X×Y or H((x, y), F˜ ) ≤ H on X×Y. By Theorem 2, we haveH(x, y, f, g)≤H,H(x, y, g, f)≤H onX×Y, or H(x, y, f, g)˜ ≤H andH(x, y, g, f˜ )≤HonX×Y. We use now Theorem 10 to conclude that f andg areL-BLD mappings for some L >0.

Remark 2. IfX, Z are domains inRⁿand Y, W are domains inR^m, then conditions (24), (25), (28), and (29) from Theorems 10 and 11 hold for instance if there existx∈X such thatf is dierentiable atxandJ_f(x)6= 0andy ∈Y such thatg is dierentiable at y and Jg(y)6= 0.

Acknowledgement. The autor was supported by Contract 2-CEx06-10/25.07.2006.

REFERENCES

[1] P. Caraman, Homeomorsme cvasiconforme n-dimensionale. Ed. Acad. Române, Bu- cure³ti, 1968. English translation: n-Dimensional Quasiconformal Mappings. Abacus Press, Tunbridge Wells, Kent, England, 1974.

[2] M. Cristea, Mappings of nite distortion and arbitrary Jacobian sign. Complex Var.

Elliptic Equ. 52 (2007), 1, 4357.

[3] M. Cristea, Quasiregularity in metric spaces. Rev. Roumaine Math. Pures Appl. 51 (2006), 291310.

[4] J. Heinonen, Lectures on Analysis on Metric Spaces. Springer, 2001.

[5] A.P. Karmazin, Ob uslovijah kvazikonformnosti prjamyh proizvedenij gomeomorzmov.

In: Extremal problemy teorii funksii, pp. 2232. Tomskij Gos. Universitet, Tomsk, 1979.

[6] O. Martio and J. Väisälä, Elliptic equations and maps on bounded length distortion.

Math. Ann. 282 (1988), 423443.

[7] O. Martio, S. Rickman and J. Väisälä, Denitions for quasiregular mappiungs. Ann.

Acad. Sci. Fenn. Ser A I, No. 448, 1969, 40 pp.

[8] Yu.G. Reshetnyak, Space Mappings of Bounded Distortion. Amer. Math. Soc., Provi- dence, RI, 1989.

[9] Yu.G. Reshetnyak, Bounds of moduli of continuity for certain mappings. Sibirsk. Mat.

Z. 7 (1966), 11061114. (Russian)

[10] S. Rickman, Quasiregularity Mappings. Springer, 1993.

[11] Elena Rusu, Direct products of quasiregular mappings. Complex Variables Theory Appl.

41 (2000), 359369.

[12] J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings. Lecture Notes in Math. 229. Springer-Verlag, BerlinHeidelbergNew York, 1971.

Received 29 May 2007 University of Bucharest

Faculty of Mathematics and Computer Science Str. Academiei 14

010014 Bucharest, Romania mcristea@fmi.unibuc.ro